Harmonic Analysis of Translation Invariant Valuations

Semyon Alesker, Andreas Bernig and Franz E. Schuster

Abstract. The decomposition of the space of continuous and translationinvariant valuations into a sum of SO(n) irreducible subspaces is obtained.A reformulation of this result in terms of a Hadwiger type theorem forcontinuous translation invariant and SO(n)-equivariant tensor valuations isalso given. As an application, symmetry properties of rigid motion invariantand homogeneous bivaluations are established and then used to prove newinequalities of Brunn–Minkowski type for convex body valued valuations.

1. Introduction and statement of main results

Let V be an n-dimensional Euclidean vector space and let A be an abeliansemigroup. A function φ defined on convex bodies (compact convex sets) inV and taking values in A is called a valuation, or additive, if

φ(K) + φ(L) = φ(K ∪ L) + φ(K ∩ L)

whenever K,L and K ∪ L are convex.The most important cases are A = R or C (scalar valued valuations),

A = SymkV (tensor valuations) and A = Kn, the semigroup of convex bodiesin V with the Minkowski addition (Minkowski valuations).

Scalar valued valuations play an important role in integral geometry.Hadwiger characterized in [26] the continuous Euclidean motion invariantvaluations. Almost all classical integral-geometric formulas can be reducedto this landmark result. For generalizations of this idea in different directions,we refer to [2, 4, 11, 15, 17, 30, 36, 46].

Tensor valuations were studied by McMullen [47], the first author [3] andLudwig [32]. Recently, a full set of kinematic formulas for tensor valuationswas obtained by Hug, Schneider and R. Schuster [27, 28].

The best known example of a Minkowski valuation is the projection body.This central notion from affine geometry has many applications in severalareas such as geometric tomography, stereology, computational geometry,optimization or functional analysis. For a systematic study of Minkowskivaluations, we refer to [22, 23, 29, 31–34, 50] and the references therein.

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In this article, we contribute to these three different directions in thetheory of valuations. Our main result may be stated in the language of scalarvalued valuations or in the language of tensor valuations. For simplicity, weassume throughout this paper that n ≥ 3, even if most of the results alsohold true for n = 2.

A valuation φ is called translation invariant if φ(K + x) = φ(K) forall x ∈ V and K ∈ Kn and φ is said to have degree i if φ(tK) = tiφ(K)for all K ∈ Kn and t > 0. We call φ even if φ(−K) = φ(K) and odd ifφ(−K) = −φ(K) for all K ∈ Kn. We denote by Val the vector space ofall continuous translation invariant complex valued valuations and we writeVal±i for its subspace of all valuations of degree i and even/odd parity. Animportant result by McMullen [45] is that

Val =⊕

0≤i≤n

(Val+i ⊕Val−i ). (1.1)

In order to state our main theorem, we need the following basic factfrom the representation theory of the group SO(n): The isomorphism classesof irreducible representations of SO(n) are parametrized by their highestweights, namely sequences of integers (λ1, λ2, . . . , λbn/2c) such that

λ1 ≥ λ2 ≥ . . . ≥ λbn/2c ≥ 0 for odd n,λ1 ≥ λ2 ≥ . . . ≥ λn/2−1 ≥ |λn/2| for even n.

(See Section 3 for the background material from representation theory.)The natural action of the group SO(n) on the space Val is given by

(ϑφ)(K) = φ(ϑ−1K), ϑ ∈ SO(n), φ ∈ Val.

Our main theorem is the following decomposition of the space Val intoirreducible SO(n)-modules.

Theorem 1. Let 0 ≤ i ≤ n. The space Vali is the direct sum of theirreducible representations of SO(n) with highest weights (λ1, . . . , λbn/2c)precisely satisfying the following additional conditions:

(i) λj = 0 for j > mini, n− i;(ii) |λj| 6= 1 for 1 ≤ j ≤ bn/2c;(iii) |λ2| ≤ 2.

In particular, under the action of SO(n) the space Vali is multiplicity free.

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Earlier versions of Theorem 1 for even valuations were obtained in [4]and [9]. These results were subsequently applied in the construction of newalgebraic structures on the space Val (see [6, 14]) which provided the meansfor a fuller understanding of the integral geometry of compact groups actingtransitively on the unit sphere (see e.g. [5, 11, 15, 17]).

For the proof of Theorem 1 we draw on methods from representationtheory, differential geometry and geometric measure theory. To be morespecific, we use a representation of smooth translation invariant valuationsvia integral currents first obtained in [7] and later refined in [13] and [12] aswell as an analysis of the action of SO(n) on the space of translation invariantdifferential forms on a contact manifold (see Sections 4 and 5).

We now state a reformulation of Theorem 1 in the language of tensorvaluations. Let (Γ, %) be a (finite dimensional, complex) representation ofSO(n). A continuous translation invariant valuation with values in Γ is calledSO(n) equivariant if

φ(ϑK) = %(ϑ)φ(K)

for all ϑ ∈ SO(n) and K ∈ Kn.

Theorem 1′. Let (Γ, %) be an irreducible SO(n) representation and let0 ≤ i ≤ n. There exists a non-trivial continuous translation invariantand SO(n) equivariant valuation of degree i with values in Γ if and onlyif the highest weight of Γ satisfies the conditions (i)-(iii) from Theorem 1.This valuation is unique up to scaling.

Since a finite dimensional representation of SO(n) can be decomposedinto a sum of irreducible representations, Theorem 1′ can be used to studythe space of equivariant Γ-valued valuations also for reducible Γ (comparethe examples in Section 5).

The case of symmetric tensors, namely Γ = SymkV , has been intensivelytreated in [2, 27, 28, 32, 47]. In these papers, translation invariance isreplaced by the more general isometry covariance. In the recent article [28],Hug, Schneider and R. Schuster explicitly determined the dimension of thespace of all continuous isometry covariant tensor valuations of a fixed rankand of a given degree of homogeneity. However, these computations do notseem to give a basis of the subspace of translation invariant tensor valuations.For the general, non-symmetric, case, not much seems to be known exceptthe construction of ΛkV ⊗ΛkV -valued translation invariant valuations in [10].

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Definition. A map ϕ : Kn ×Kn → C is called a bivaluation if ϕ is additivein each argument. A bivaluation ϕ is called translation biinvariant if ϕ isinvariant under independent translations of its arguments and ϕ is said tohave bidegree (i, j) if ϕ(tK, sL) = tisjϕ(K,L) for all K,L ∈ Kn and t, s > 0.We say ϕ is O(n) invariant (resp. SO(n) invariant) if ϕ(ϑK, ϑL) = ϕ(K,L)for all K,L ∈ Kn and ϑ ∈ O(n) (resp. ϑ ∈ SO(n)).

In their book on geometric probability, Klain and Rota [30] pose theproblem to classify all ”invariant” bivaluations. First such classificationresults were obtained recently by Ludwig [35]. In Section 6 we obtain adescription of all continuous translation biinvariant bivaluations which canbe seen as a starting point for systematic investigations of this problem.

As an application of Theorem 1, we obtain the following importantsymmetry property of rigid motion invariant homogeneous bivaluations.

Theorem 2. If ϕ : Kn×Kn → R is a continuous translation biinvariant andO(n) invariant bivaluation of bidegree (i, i), 0 ≤ i ≤ n, then

ϕ(K,L) = ϕ(L,K) (1.2)

for every K,L ∈ Kn.

As a byproduct of our proof of Theorem 2, we also obtain that if thebivaluation ϕ is as above but merely SO(n) invariant, then (1.2) still holdstrue if (i, n) 6= (2k + 1, 4k + 2), k ∈ N. If n ≡ 2 mod 4, then there existcontinuous translation biinvariant and SO(n) invariant bivaluations ofbidegree

(n2, n

2

)which are not symmetric.

The symmetry property established in Theorem 1 in combination withtechniques developed by Lutwak [37, 41] can be used to obtain geometricinequalities for Minkowski valuations. Recall that a map Φ : Kn → Kn

is called a Minkowski valuation if Φ is additive with respect to the usualMinkowski addition of convex sets. We denote by MVali the set of allcontinuous translation invariant Minkowski valuations of degree i.

A convex body K is uniquely determined by its support functionh(K, u) = maxu · x : x ∈ K, for u ∈ Sn−1. Among the most importantexamples of Minkowski valuations is the projection operator Π ∈ MValn−1:The projection body ΠK of K is the convex body defined by

h(ΠK, u) = voln−1(K|u⊥), u ∈ Sn−1,

whereK|u⊥ denotes the projection ofK onto the hyperplane orthogonal to u.For the special role of the map Π in the theory of valuations we refer to [31].

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Let 0 ≤ i ≤ n−1. If Φi ∈ MVali is O(n) equivariant, i.e. Φi(ϑK) = ϑΦiKfor all K ∈ Kn and ϑ ∈ O(n), then the map

ϕ(K,L) = V (ΦiK,L[i], B[n− i− 1]), K, L ∈ Kn,

where V (ΦiK,L[i], B[n− i− 1]) denotes the mixed volume of ΦiK, i copiesof L and n − i − 1 copies of the Euclidean unit ball B, is a translationbiinvariant and O(n) invariant bivaluation of bidegree (i, i). By Theorem 2,it is symmetric in K and L.

In the particular case where i = n − 1 and Φn−1 = Π, this symmetryproperty is well known. Its variants and generalizations have been usedextensively for establishing geometric inequalities related to convex and starbody valued valuations (see [19, 21, 24, 37, 38, 40–44, 50] and Section 7).Complex versions of the projection body were recently studied in [1], theysatisfy similar symmetry properties.

In the following we give one example of the type of inequalities that canbe derived from Theorem 2. To this end let us recall a version of the classicalBrunn–Minkowski inequality. For i ∈ 0, . . . , n, let Vi(K) denote the i-thintrinsic volume of K ∈ Kn. The Brunn–Minkowski inequality for intrinsicvolumes states the following: If 2 ≤ i ≤ n and K,L ∈ Kn have non-emptyinterior, then

Vi(K + L)1/i ≥ Vi(K)1/i + Vi(L)1/i, (1.3)

with equality if and only if K and L are homothetic.In [37, 41] Lutwak obtained inequalities of Brunn–Minkowski type for a

well known family of Minkowski valuations derived from the projection bodyoperator. As an application of Theorem 2, we show that inequalities (1.3)and Lutwak’s inequalities for derived projection operators of order i are infact part of a larger family of Brunn–Minkowski type inequalities which holdfor all continuous translation invariant and SO(n) equivariant Minkowskivaluations of a given degree.

Theorem 3. Suppose that Φi ∈ MVali, 1 ≤ i ≤ n−1, is SO(n) equivariant.If K,L ∈ Kn have non-empty interior, then

Vi+1(Φi(K + L))1/i(i+1) ≥ Vi+1(ΦiK)1/i(i+1) + Vi+1(ΦiL)1/i(i+1).

If i ≥ 2 and Φi maps convex bodies with non-empty interior to bodies withnon-empty interior, then equality holds if and only if K and L are homothetic.

The special case of Theorem 3 for even Minkowski valuations was recentlyestablished by other methods by the third author [50].

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2. Translation invariant valuations

In the following we collect background results on translation invariantcomplex valued valuations needed in subsequent sections. In particular, werecall the definition of O(n) finite valuations and smooth valuations as wellas their representation via integral currents.

A classical theorem of Minkowski states that the volume of a Minkowskilinear combination t1K1 + . . . + tkKk of convex bodies K1, . . . , Kk can beexpressed as a homogeneous polynomial of degree n:

Vn(t1K1 + . . .+ tkKk) =k∑

i1,...,in=1

V (Ki1 , . . . , Kin)ti1 · · · tin .

The coefficients V (Ki1 , . . . , Kin) are called mixed volumes of Ki1 , . . . , Kin .Clearly, V (K, . . . ,K) = Vn(K). Moreover, mixed volumes are symmetric,non-negative and multilinear with respect to Minkowski linear combinations.They are also continuous with respect to the Hausdorff metric and satisfythe following two properties:

• If K,L ∈ Kn such that K ∪ L ∈ Kn, and C = (K1, . . . , Ki), then

Vi(K,C) + Vi(L,C) = Vi(K ∪ L,C) + Vi(K ∩ L,C),

where Vi(K,C) denotes the mixed volume V (K, . . . ,K,K1, . . . , Ki).

• Mixed volumes are invariant under independent translations of theirarguments and they are invariant under simultaneous unimodular lineartransformations, i.e., if K1, . . . , Kn ∈ Kn and A ∈ SL(n), then

V (AK1, . . . , AKn) = V (K1, . . . , Kn).

Recall that we denote by Val the vector space of continuous translationinvariant complex valued valuations and we write Val±i for its subspaces ofall valuations of degree i and even/odd parity.

It is easy to see that the space Val0 is one-dimensional. The analogous(non-trivial) statement for Valn was proved by Hadwiger [26, p. 79].

The following consequence of McMullen’s decomposition (1.1) is wellknown.

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Corollary 2.1. Let C ∈ Kn be a fixed convex body with non-empty interior.The space Val becomes a Banach space under the norm

‖φ‖ = sup|φ(K)| : K ⊆ C.

Moreover, a different choice of C yields an equivalent norm.

The natural continuous action of the group GL(n) on the Banach spaceVal is given by

(Aφ)(K) = φ(A−1K), A ∈ GL(n), φ ∈ Val.

Note that the subspaces Val±i ⊆ Val are invariant under this GL(n) action.The following result is known as the Irreducibility Theorem. It implies a

conjecture by McMullen that the linear combinations of mixed volumes forma dense subspace in Val.

Theorem 2.2. ([4]) The natural action of GL(n) on Val±i is irreducible forevery i ∈ 0, . . . , n.

In the following it will be important for us to work with two differentdense subsets of valuations in Val:

Definition A valuation φ ∈ Val is called O(n) finite if the O(n) orbit of φ,i.e. the subspace spanϑφ : ϑ ∈ O(n), is finite dimensional.

A valuation φ ∈ Val is called smooth if the map GL(n) → Val defined byA 7→ Aφ is infinitely differentiable.

The notions of O(n) finite and smooth valuations are special cases of moregeneral well known concepts from representation theory (see e.g. [51]).

We denote the space of continuous translation invariant and O(n) finitevaluations by Valf and we write Val∞ for the space of smooth translationinvariant valuations. For the subspaces of homogeneous valuations of givenparity in Valf and Val∞ we write Val±,f

i and Val±,∞i , respectively.

It is well known (cf. [16, p. 141] and [51, p. 32]) that the set of O(n) finitevaluations Val±,f

i is a dense O(n) invariant subspace of Val±i and that the setof smooth valuations Val±,∞

i is a dense GL(n) invariant subspace of Val±i .Moreover, Valf ⊆ Val∞ and from (1.1) one easily deduces that the spacesValf and Val∞ admit direct sum decompositions into their correspondingsubspaces of homogeneous valuations of given parity.

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An equivalent description of smooth valuations can be given in terms ofthe normal cycle map. Let SV = V × Sn−1 denote the unit sphere bundleon V . For K ∈ Kn and x ∈ ∂K, we write N(K, x) for the normal cone of Kat x. The normal cycle (or generalized normal bundle) of a convex body Kis the Lipschitz submanifold of SV defined by

nc(K) = (x, u) ∈ SV : x ∈ ∂K, u ∈ N(K, x).For 0 ≤ i ≤ n − 1, let Ωi,n−i−1 denote the space of smooth translation

invariant differential forms of bidegree (i, n − i − 1) on SV . The followingresult is a special case of [7, Theorem 5.2.1]:

Lemma 2.3. If 0 ≤ i ≤ n−1, then the map ν : Ωi,n−i−1 → Val∞i , defined by

ν(ω)(K) =

∫nc(K)

ω, (2.1)

is surjective.

The kernel of the map ν was described in [13] in terms of the Ruminoperator [48], a second order differential operator which acts on smooth formson the sphere bundle. A refined version of this result (stated in Section 4as Theorem 4.3) was recently proved in [12] and will be crucial in the proofof Theorem 1. We also remark that recently a broader notion of smoothvaluations in the setting of smooth manifolds was introduced, see [7]. Theclassical concept of valuations as used in this article is in some sense aninfinitesimal version of this more general notion.

The description of smooth valuations provided by Lemma 2.3 was themain tool used in [13] to establish a Hard Lefschetz Theorem for translationinvariant valuations (see also [5, 8]). The next statement is an immediateconsequence of this result:

Theorem 2.4. For every i ∈ 0, . . . , n, the spaces Val∞i and Val∞n−i areisomorphic as SO(n) modules.

3. Irreducible representations of SO(n) and O(n)

In this section we recall some well known results concerning irreduciblerepresentations of the groups SO(n) and O(n), n ≥ 3. As a general referencefor this material we recommend the books by Brocker and tom Dieck [16],Fulton and Harris [18], and Goodman and Wallach [20].

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Since SO(n) and O(n) are compact Lie groups, all their irreduciblerepresentations are finite dimensional. The equivalence classes of irreduciblecomplex representations of SO(n) are indexed by their highest weights,namely bn/2c-tuples of integers (λ1, λ2, . . . , λbn/2c) such that

λ1 ≥ λ2 ≥ . . . ≥ λbn/2c ≥ 0 for odd n,λ1 ≥ λ2 ≥ . . . ≥ λn/2−1 ≥ |λn/2| for even n.

(3.1)

In the following we use Γλ to denote any isomorphic copy of an irreduciblerepresentation of SO(n) with highest weight λ = (λ1, λ2, . . . , λbn/2c).

Examples:

(a) The only one dimensional (complex) representation of SO(n) is thetrivial representation; it corresponds to the SO(n) module Γ(0,...,0).

(b) We denote by VC = V ⊗ C the complexification of V . The standardrepresentation of SO(n) on VC corresponds to Γ(1,0,...,0).

(c) For every 0 ≤ i ≤ bn/2c − 1, the exterior power ΛiVC is an irreducibleSO(n) module with λ = (1, . . . , 1, 0, . . . , 0), where 1 appears i times.

If n = 2k + 1 is odd, the exterior power ΛkVC is also irreducible; but ifn = 2k is even, it splits as ΛkVC = Γ(1,...,1) ⊕ Γ(1,...,1,−1)

For every i ∈ 0, . . . , n, there is a natural isomorphism of SO(n)modules

ΛiVC ∼= Λn−iVC. (3.2)

(d) For k ≥ 2, the symmetric power SymkVC is not irreducible as SO(n)module; its decomposition into irreducible submodules is given by

SymkVC =

bk/2c⊕j=0

Γ(k−2j,0,...,0). (3.3)

A description of the irreducible representations of the full orthogonalgroup O(n) can be given in terms of the irreducible representations of itsidentity component SO(n) (cf. [20, p. 249]). The main difference arises fromthe fact that O(n) has a non-trivial one dimensional representation, calledthe determinant representation, which corresponds to the O(n) module ΛnVC.

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Lemma 3.1. Let λ = (λ1, . . . , λbn/2c) be a tuple of integers satisfying (3.1).

(a) If n is odd, then the irreducible representation Γλ of SO(n) is therestriction of two non-isomorphic irreducible O(n) representations Γλ

and Γλ ⊗ ΛnVC.

(b) If n is even and λn/2 = 0, then the irreducible representation Γλ

of SO(n) is the restriction of two non-isomorphic irreducible O(n)representations Γλ and Γλ ⊗ ΛnVC. If λn/2 6= 0, then Γλ is not such arestriction.

(c) If n is even and λn/2 6= 0, then the SO(n) representation Γλ ⊕ Γλ′,where λ′ := (λ1, . . . , λn/2−1,−λn/2), is the restriction of an irreducibleO(n) representation Γλ.

Moreover, all irreducible representations of O(n) are determined in this way.

Let Γ be a (not necessarily irreducible) complex SO(n) or O(n) module.Recall that the dual representation is defined on the dual space Γ∗ by

(ϑu∗)(v) = u∗(ϑ−1v), ϑ ∈ SO(n), u∗ ∈ Γ∗, v ∈ Γ.

We say that Γ is self-dual if Γ and Γ∗ are isomorphic representations. Themodule Γ is called real if there exists a non-degenerate symmetric SO(n)invariant, or O(n) respectively, bilinear form on Γ. In particular, if Γ is real,then Γ is also self-dual.

The following lemma (cf. [16, p. 292]) will be critical in the proof ofTheorem 2:

Lemma 3.2. Let λ = (λ1, . . . , λbn/2c) be a tuple of integers satisfying (3.1).

(a) If n 6≡ 2 mod 4, then all representations of SO(n) are real.

(b) If n ≡ 2 mod 4, then the irreducible representation Γλ of SO(n) is realif and only if λn/2 = 0. If λn/2 6= 0, then the dual of Γλ is Γλ′.

Moreover, all representations of O(n) are real.

An essential tool in the classification of irreducible modules of a compactgroup is the character of a representation: Let Γ be a finite dimensional(complex) SO(n) module and let % : SO(n) → GL(Γ) be the corresponding

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representation. The character of Γ is the function char Γ : SO(n) → Cdefined by

(char Γ)(ϑ) = Tr %(ϑ),

where Tr %(ϑ) is the trace of the linear map %(ϑ) : Γ → Γ.A complex representation is determined up to isomorphism by its

character. Moreover, from properties of the trace map, one immediatelyobtains several useful arithmetic properties of characters: If Γ and Θ arefinite dimensional SO(n) modules, then

char(Γ⊕Θ) = char Γ + char Θ (3.4)

andchar(Γ⊗Θ) = char Γ · char Θ. (3.5)

The character of the irreducible SO(n) modules Γλ with highest weightsλ = (λ1, . . . , λbn/2c) are described by Weyl’s character formula. However,more important for us is a consequence of this description, known as thesecond determinantal formula, which we describe in the following.

Let λ = (λ1, . . . , λbn/2c) be a tuple of non-negative integers satisfying (3.1).We define the SO(n) module Γλ by

Γλ :=

Γλ ⊕ Γλ′ if n is even and λn/2 6= 0,Γλ otherwise.

The second determinantal formula expresses char Γλ as a polynomial in thecharacters Ei of the fundamental representations ΛiVC, i ∈ Z. (Note thatE0 = En = 1 and that we use the convention Ei = 0 for i < 0 or i > n.)

Given a tuple of non-negative integers λ = (λ1, . . . , λbn/2c) satisfying (3.1),recall that the conjugate of λ is the s := λ1 tuple µ = (µ1, . . . , µs) defined bysaying that µj is the number of terms in λ that are greater than or equal j.The second determinantal formula (cf. [18, p. 409]) can be stated as follows:

Theorem 3.3. Let λ = (λ1, . . . , λbn/2c) be a tuple of non-negative integerssatisfying (3.1) and let µ = (µ1, . . . , µs) be the conjugate of λ. The characterof Γλ equals the determinant of the s× s-matrix whose i-th row is given by(

Eµi−i+1 Eµi−i+2 + Eµi−i · · · Eµi−i+s + Eµi−i−s+2

). (3.6)

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It is sometimes convenient for us to take s > λ1 in the definition ofthe conjugate of λ. This just introduces additional zeros at the end of theconjugate tuple. However, note that this does not change the determinantof the matrix defined by (3.6).

In the following we use #(λ, j) to denote the number of terms in atuple of (non-negative) integers λ = (λ1, . . . , λbn/2c) which are equal to j.As a consequence of Theorem 3.3, we note the following auxiliary resultwhich will be needed in the proof of Theorem 1.

Corollary 3.4. If i, j ∈ N are such that n/2 ≤ i ≤ n and i+ j ≤ n, then

EiEj − Ei−1Ej−1 =∑

λ

char Γλ, (3.7)

where the sum ranges over all bn/2c-tuples of non-negative integersλ = (λ1, . . . , λbn/2c) satisfying (3.1) and

λ1 ≤ 2, #(λ, 1) = n− i− j, #(λ, 2) ≤ j. (3.8)

Proof : If λ = (λ1, . . . , λbn/2c) is a tuple of non-negative integers satisfying(3.1) and (3.8), then the conjugate of λ is given by µ = (µ1, µ2), whereµ2 = #(λ, 2) ≤ j and µ1 − µ2 = #(λ, 1) = n− i− j. Thus, by Theorem 3.3,the character of Γλ is given by

char Γλ = det

(Eµ2+k Eµ2+k+1 + Eµ2+k−1

Eµ2−1 Eµ2 + Eµ2−2

),

where k = n− i− j. Consequently, the right hand side of (3.7) is

∑λ

char Γλ =

j∑µ2=0

(Eµ2+k(Eµ2 + Eµ2−2)− Eµ2−1(Eµ2+k+1 + Eµ2+k−1)

)= En−iEj − En−(i−1)Ej−1.

To finish the proof, note that En−i = Ei by (3.2).

An important class of (infinite dimensional) representations of a Lie groupG are those induced from closed subgroups H of G. Although in this articlewe will only need the case G = SO(n) and H = SO(n− 1), we shall explainthis construction for a general compact Lie group G and its closed subgroupH. To this end, for any finite dimensional complex vector space Γ, we denoteby C∞(G; Γ) the space of all smooth functions f : G→ Γ.

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If Θ is any representation of G, clearly we obtain a representation ResGHΘ

of H by restriction. Conversely, each H module Γ induces a representationof G as follows: Let IndG

HΓ ⊆ C∞(G; Γ) be the space of functions defined by

IndGHΓ :=

f ∈ C∞(G; Γ) : f(gh) = h−1f(g) for all g ∈ G, h ∈ H

.

The (smooth) induced representation of G on IndGHΓ is now given by left

translation(gf)(u) = f(g−1u), g, u ∈ G.

A basic result on induced representations is the well known FrobeniusReciprocity Theorem (cf. [20, p. 523]):

Theorem 3.5. If Θ is a G module and Γ is an H module, then there is acanonical vector space isomorphism

HomG(Θ, IndGHΓ) ∼= HomH(ResG

HΘ,Γ).

Here, HomG denotes the space of continuous linear G equivariant maps.Recall that if Θ is an irreducible G module, by Schur’s lemma, the

multiplicity m(Ξ,Θ) of Θ in an arbitrary G module Ξ is given by

m(Ξ,Θ) = dim HomG(Ξ,Θ) = dim HomG(Θ,Ξ).

Thus, by the Frobenius Reciprocity Theorem, if Θ and Γ are irreducible,then the multiplicity of Θ in IndG

HΓ equals the multiplicity of Γ in ResGHΘ.

In order to apply Theorem 3.5 in our situation, where G = SO(n) and

H = SO(n − 1), we will need a formula for decomposing ResSO(n)SO(n−1)Γ into

irreducible SO(n−1) modules. This is the content of the following branchingtheorem (cf. [18, p. 426]):

Theorem 3.6. If Γλ, with λ = (λ1, . . . , λbn/2c) satisfying (3.1), is anirreducible representation of SO(n), then

ResSO(n)SO(n−1)Γλ =

⊕µ

Γµ, (3.9)

where the sum ranges over all µ = (µ1, . . . , µk) with k := b(n− 1)/2c andλ1 ≥ µ1 ≥ λ2 ≥ µ2 ≥ . . . ≥ µk−1 ≥ λbn/2c ≥ |µk| for odd n,λ1 ≥ µ1 ≥ λ2 ≥ µ2 ≥ . . . ≥ µk ≥ |λn/2| for even n.

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4. The Rumin–de Rham complex

We state in this section a refinement of the description of translationinvariant smooth valuations via integral currents. We also establish anauxiliary result which will enable us to subsequently employ the machineryfrom representation theory explained in Section 3.

Recall that SV = V × Sn−1 denotes the unit sphere bundle. The naturalsmooth (left) action of SO(n) on SV is given by

lϑ(x, u) := (ϑx, ϑu), ϑ ∈ SO(n), (x, u) ∈ SV. (4.1)

Similarly, each y ∈ V determines a smooth map ty : SV → SV by

ty(x, u) = (x+ y, u), (x, u) ∈ SV. (4.2)

The canonical contact form α on SV is the one form defined by

α|(x,u)(w) = 〈u, d(x,u)π(w)〉, w ∈ T(x,u)SV,

where π : SV → V denotes the canonical projection and d(x,u)π its differentialat (x, u) ∈ SV . In this way, SV becomes a 2n − 1 dimensional contactmanifold. The kernel of α defines the contact distribution Q := kerα. Therestriction of dα to Q is a non-degenerate two form. In this way, each Q(x,u)

becomes a symplectic vector space.The Reeb vector field R on SV is defined by R(x,u) = (u, 0). It is the

unique vector field on SV such that α(R) = 1 and iRdα = 0, where iRdαdenotes the interior product of R and dα. At each point (x, u), Q(x,u) isthe orthogonal sum of two copies of TuS

n−1 and, consequently, we have anorthogonal splitting of the tangent space T(x,u)SV given by

T(x,u)SV = spanRR(x,u) ⊕ TuSn−1 ⊕ TuS

n−1. (4.3)

The product structure of SV induces a bigrading on the vector spaceΩ∗(SV ) of complex valued smooth differential forms given by

Ω∗(SV ) =⊕i,j

Ωi,j(SV ),

where Ωi,j(SV ) denotes the subspace of Ω∗(SV ) of forms of bidegree (i, j).We write Ωi,j ⊆ Ωi,j(SV ) for the subspace of translation invariant forms, i.e.,

Ωi,j = ω ∈ Ωi,j(SV ) : t∗yω = ω for all y ∈ V .

14

Here, t∗y is the pullback of the map ty : SV → SV defined in (4.2). Note thatthe restriction of the exterior derivative d to Ωi,j has bidegree (0, 1).

The vector space Ωi,j becomes an SO(n) module under the (continuous)action

ϑω = l∗ϑ−1ω, ϑ ∈ SO(n), ω ∈ Ωi,j.

An important SO(n) submodule of Ωi,j is given by the space Ωi,jv of vertical

forms, defined byΩi,j

v := ω ∈ Ωi,j : α ∧ ω = 0.

Note that a differential form ω ∈ Ωi,j is vertical if and only if it vanishes onthe contact distribution Q of SV .

The SO(n) submodule Ωi,jh ⊆ Ωi,j of horizontal forms, is given by

Ωi,jh := ω ∈ Ωi,j : iRω = 0 ∼= Ωi,j/Ωi,j

v .

It follows from (4.3) and the definition of Ωi,jh that ω ∈ Ωi,j is horizontal if

and only ifω|(x,u) ∈ ΛiT ∗uS

n−1 ⊗ ΛjT ∗uSn−1 ⊗ C

for every x ∈ V and each u ∈ Sn−1. In the following we will therefore simplywrite ω|u instead of ω|(x,u) whenever ω ∈ Ωi,j

h and (x, u) ∈ SV .We now fix a point u0 ∈ Sn−1 and let SO(n−1) be the stabilizer of SO(n)

at u0. For u ∈ Sn−1, we denote by Wu := TuSn−1 ⊗ C the complexification

of the tangent space TuSn−1 and we write W0 to denote Wu0 .

Lemma 4.1. For i, j ∈ N, there is an isomorphism of SO(n) modules

Ωi,jh∼= Ind

SO(n)SO(n−1)(Λ

iW ∗0 ⊗ ΛjW ∗

0 ).

Proof : First note that, for each ϑ ∈ SO(n), the differential of the maplϑ : SV → SV defined in (4.1) induces a linear isomorphism

du0lϑ := (du0lϑ)∗ : ΛiW ∗ϑu0

⊗ ΛjW ∗ϑu0

→ ΛiW ∗0 ⊗ ΛjW ∗

0 .

Moreover, the natural representation of the group SO(n − 1) on the space

ΛiW ∗0 ⊗ ΛjW ∗

0 is given by η 7→ du0lη−1 .

Suppose now that ω ∈ Ωi,jh . We define fω : SO(n) → ΛiW ∗

0 ⊗ ΛjW ∗0 by

fω(ϑ) = du0lϑ(ω|ϑu0).

15

Clearly, we have fω(ϑη) = η−1fω(ϑ) for every ϑ ∈ SO(n) and η ∈ SO(n− 1).

This shows that fω ∈ IndSO(n)SO(n−1)(Λ

iW ∗0 ⊗ ΛjW ∗

0 ).

Conversely, let f ∈ IndSO(n)SO(n−1)(Λ

iW ∗0 ⊗ ΛjW ∗

0 ). We define a horizontal

form ωf ∈ Ωi,jh by

ωf |ϑu0 = du0lϑ−1

(f(ϑ)).

It is not difficult to show that ω is well defined, i.e. if ϑu0 = ϑ′u0 for someϑ, ϑ′ ∈ SO(n), then ω|ϑu0 = ω|ϑ′u0 .

The observation that the SO(n) equivariant linear maps ω 7→ fω andf 7→ ωf are inverse to each other finishes the proof.

Let I i,j denote the SO(n) invariant subspace of Ωi,j defined by

I i,j := ω ∈ Ωi,j : ω = α ∧ ξ + dα ∧ ψ, ξ ∈ Ωi−1,j, ψ ∈ Ωi−1,j−1.

Finally, we denote by Ωi,jp the SO(n) module of primitive forms defined

as the quotientΩi,j

p := Ωi,j/I i,j. (4.4)

An equivalent description of primitive forms can be given as follows: Themultiplication by the symplectic form −dα induces an SO(n) equivariantlinear operator L : Ωi,j

h → Ωi+1,j+1h which is injective if i + j ≤ n − 2.

Moreover, it follows from the definition of Ωi,jp that in this case

Ωi,jp = Ωi,j

h /LΩi−1,j−1h . (4.5)

From Lemma 4.1 and (4.5), we now immediately obtain

Corollary 4.2. If i, j ∈ N are such that i + j ≤ n − 1, then there is anisomorphism of SO(n) modules

Ωi,jp ⊕ Ind

SO(n)SO(n−1)(Λ

i−1W ∗0 ⊗ Λj−1W ∗

0 ) ∼= IndSO(n)SO(n−1)(Λ

iW ∗0 ⊗ ΛjW ∗

0 ).

Primitive forms are of particular importance for us since the space Val∞ifits into an exact sequence of the spaces Ωi,j

p , as was recently establishedin [12]. In order to describe this sequence, note that dI i,j ⊆ I i,j+1. Thus,by definition (4.4), the exterior derivative, on one hand, induces a linearoperator dQ : Ωi,j

p → Ωi,j+1p and, on the other hand, integration over the

normal cycle induces a linear map ν : Ωi,n−i−1p → Val∞i (cf. Lemma 2.3).

Clearly, both operators are SO(n) equivariant.

16

Theorem 4.3. If 0 ≤ i ≤ n, then there is an exact SO(n) equivariantsequence of SO(n) modules

0 → ΛiVC → Ωi,0p

dQ→ Ωi,1p

dQ→ . . .dQ→ Ωi,n−i−1

pν→ Val∞i → 0.

5. Proof of Theorems 1 and 1′

Theorems 1 and 1′ are just reformulations of each other. We give theproof of Theorem 1 first and then show how Theorem 1′ follows from it.

Proof of Theorem 1 : The cases i = 0 and i = n are trivial. Moreover, byTheorem 2.4, we may assume that n/2 ≤ i < n.

Let Γλ be an arbitrary irreducible SO(n) module of highest weightλ = (λ1, . . . , λbn/2c). It is well known (and a consequence of Corollary 4.2)that the multiplicity of Γλ in the spaces Ωi,j

p of primitive forms is finite. Thesame holds true for the spaces Vali since they are quotients of Ωi,n−i−1

p byTheorem 4.3. Thus, by Theorem 4.3, we have

m(Vali, λ) = (−1)n−im(ΛiVC, λ) +n−i−1∑

j=0

(−1)n−1−i−jm(Ωi,jp , λ), (5.1)

wherem( · , λ) denotes the multiplicity of Γλ in the respective SO(n) modules.Let W ∼= W ∗ denote the (complex) standard representation of SO(n−1).

By Corollary 4.2 and (3.4), we have

m(Ωi,jp ,λ) = m

(Ind

SO(n)SO(n−1)(Λ

iW⊗ ΛjW ),λ)−m

(Ind

SO(n)SO(n−1)(Λ

i−1W⊗ Λj−1W ),λ).

Thus, it follows from an application of Corollary 3.4 (with n replaced by n−1and 0 ≤ j ≤ n− 1− i) that

m(Ωi,jp , λ) =

∑σ

m(Ind

SO(n)SO(n−1)Γσ, λ

), (5.2)

where the sum ranges over all k := b(n−1)/2c-tuples of non-negative highestweights σ = (σ1, . . . , σk) of SO(n− 1) such that

σ1 ≤ 2, #(σ, 1) = n− 1− i− j, #(σ, 2) ≤ j.

17

If Pi denotes the union of these k-tuples of non-negative highest weightsof SO(n− 1), then, by (5.1) and (5.2),

m(Vali, λ) = (−1)n−im(ΛiVC, λ) +∑σ∈Pi

(−1)|σ|m(Ind

SO(n)SO(n−1)Γσ, λ

). (5.3)

Let λ∗ = (λ∗1, . . . , λ∗bn/2c), where λ∗1 := minλ1, 2 and λ∗j := |λj| for every

1 < j ≤ bn/2c. By Theorem 3.5, Theorem 3.6 and the definition of Γσ, wehave ∑

σ∈Pi

(−1)|σ|m(Ind

SO(n)SO(n−1)Γσ, λ

)=

∑µ

(−1)|µ|,

where the sum on the right ranges over all sequences µ = (µ1, . . . , µk) withµn−i = 0 and

λ∗1 ≥ µ1 ≥ λ∗2 ≥ µ2 ≥ . . . ≥ µk−1 ≥ λ∗bn/2c ≥ |µk| for odd n,

λ∗1 ≥ µ1 ≥ λ∗2 ≥ µ2 ≥ . . . ≥ µk ≥ λ∗n/2 for even n.

If λ∗n−i+1 > 0, there is no such sequence. If λ∗n−i+1 = 0, we obtain

∑σ∈Pi

(−1)|σ|m(Ind

SO(n)SO(n−1)Γσ, λ

)=

n−i−1∏j=1

λ∗j∑µj=λ∗j+1

(−1)µj .

This product vanishes if the λ∗j , j = 1, . . . , n− i, do not all have the sameparity. If the λ∗j , j = 1, . . . , n− i, all have the same parity, the product above

equals (−1)(n−i−1)λ∗1 . Consequently, we obtain for i > n/2,

∑σ∈Pi

(−1)|σ|m(Ind

SO(n)SO(n−1)Γσ, λ

)=

(−1)n−i−1 if Γλ

∼= Λn−iVC,1 if λ satisfies (i), (ii), (iii),0 otherwise.

If i = n/2, in which case n is even, then

∑σ∈Pi

(−1)|σ|m(Ind

SO(n)SO(n−1)Γσ, λ

)=

(−1)i−1 if λ = (1, . . . , 1,±1),1 if λ satisfies (i), (ii) and (iii),0 otherwise.

Plugging this into (5.3) and using that Λn/2VC = Γ(1,...,1) ⊕ Γ(1,...,1,−1) if nis even and Λn−iVC ∼= ΛiVC for every i ∈ 0, . . . , n, completes the proof.

18

Next we explain how Theorem 1′ can be deduced from Theorem 1. Theargument presented here in fact shows that Theorem 1 and 1′ are equivalent.

Proof of Theorem 1′ : Let Γ = Γµ be an irreducible SO(n)-module. The spaceof Γ-valued valuations is isomorphic to Val⊗ Γ.

Let S denote the set of highest weights of SO(n) satisfying conditions(i)-(iii). By Theorem 1, we have

dim(Vali ⊗ Γ)SO(n) = dim(Valfi ⊗ Γ)SO(n) =∑λ∈S

dim(Γλ ⊗ Γµ)SO(n).

Here and in the following, the superscript SO(n) denotes the subspaces ofSO(n) invariant elements. The Γλ, λ ∈ S are not necessarily self dual (com-pare Lemma 3.2). However, if λ ∈ S, then also λ′ ∈ S, where λ′ is the highestweight of Γ∗λ. Thus, by Schur’s lemma, we have

dim(Vali ⊗ Γ)SO(n) =∑λ∈S

dim HomSO(n)(Γλ,Γµ) =

1 if µ ∈ S;0 otherwise.

Examples:

(a) If Γ = Γ(0,...,0)∼= C is the trivial representation, then (Val⊗ Γ)SO(n) ∼=

ValSO(n) is the vector space of all continuous rigid motion invariantvaluations and Theorem 1′ reduces to Hadwiger’s characterization ofintrinsic volumes.

(b) If Γ = Γ(1,0,...,0)∼= VC is the standard representation, then there is no

translation invariant and SO(n) equivariant continuous valuation withvalues in Γ.

(c) By (3.3), we have, for 1 ≤ i ≤ n− 1,

dim(Vali ⊗ SymkVC)SO(n) =

k/2 + 1 if k is even(k − 1)/2 if k is odd.

In particular, if k = 2, then there exist (up to constant multiples) twotranslation invariant and SO(n) equivariant continuous Sym2VC valuedvaluations of a given degree 1 ≤ i ≤ n − 1. These valuations areexplicitly known (see [27, 32, 47]).

19

(d) If Γ = Γ(1,1,0...,0) = Λ2VC is the space of skew-symmetric tensors ofrank two, then there is no translation invariant and SO(n) equivariantcontinuous valuation with values in Γ. This answers (the translationinvariant case of) a question by Yang [52].

(e) The unique translation invariant and SO(n) equivariant continuousvaluation with values in Γ(2,...,2,0,...,0) was constructed in [10].

6. Bivaluations

We turn now to the study of bivaluations. In particular, we will presentthe proof of Theorem 2 at the end of this section.

We denote the vector space of all continuous translation biinvariantcomplex valued bivaluations by BVal and we write BVali,j for its subspace ofall bivaluations of bidegree (i, j). An immediate consequence of McMullen’sdecomposition (1.1) of the vector space Val is a corresponding result for thespace BVal:

Corollary 6.1. The space BVal admits a decomposition

BVal =n⊕

i,j=0

BVali,j.

Corollary 6.1 implies an analog of Corollary 2.1 for the space of translationbiinvariant bivaluations as follows.

Corollary 6.2. Let C ∈ Kn be a fixed convex body with non-empty interior.The space BVal becomes a Banach space under the norm

‖ϕ‖ = sup|ϕ(K,L)| : K,L ⊆ C.

Moreover, a different choice of C yields an equivalent norm.

The group O(n)×O(n) acts continuously on the Banach space BVal by

((η, ϑ)φ)(K,L) = φ(η−1K,ϑ−1L), (η, ϑ) ∈ O(n)×O(n), ϕ ∈ BVal.

We denote by BValf the subspace of bivaluations with finite dimensionalO(n)×O(n) orbit. Since O(n)×O(n) is compact, BValf is a dense subspaceof BVal (see e.g. [16, p. 141]).

20

Proposition 6.3. Let 0 ≤ i, j ≤ n. The linear map ι : Valfi ⊗ Valfj →BValfi,j, induced by

ι(φ⊗ ψ)(K,L) = φ(K)ψ(L), (6.1)

is an isomorphism of O(n)×O(n) modules.

Proof : It is easy to see that the map ι is O(n) × O(n) equivariant andinjective. It remains to prove that it is onto.

It is well known that every irreducible O(n)×O(n) module is of the formΓ ⊗ Θ, where Γ,Θ are irreducible O(n) modules (c.f. [16, p. 82]). Thus, ifϕ ∈ BValfi,j belongs to a subspace isomorphic to Γ ⊗ Θ, the valuationϕ( · , L) ∈ Vali belongs to a subspace which is isomorphic to Γ as an O(n)module for every L ∈ Kn. Since any O(n) representation whose restrictionto SO(n) is multiplicity free, is itself multiplicity free, it follows fromTheorem 1 that the irreducible subspace of Vali which is isomorphic to Γhas multiplicity at most one.

If φ1, . . . , φl is a basis of the isomorphic copy of Γ in Vali, then

ϕ( · , L) =l∑

k=1

φk( · )ψk(L),

where ψk(L) are coefficients depending on L. It is not difficult to show thatψk ∈ Valj and that ψk belongs to an isomorphic copy of Θ in Valj for everyk ∈ 1, . . . , l. Thus, we have shown that ϕ is the image under the map ι ofthe element

∑lk=1 φk ⊗ ψk ∈ Valfi ⊗Valfj .

After these preparations, we are now in a position to proof the followingrefinement of Theorem 2.

Theorem 6.4. Suppose that ϕ ∈ BVali,i, where 0 ≤ i ≤ n.

(a) If ϕ is O(n) invariant, then ϕ(K,L) = ϕ(L,K) for every K,L ∈ Kn.

(b) If ϕ is SO(n) invariant and (i, n) 6= (2k + 1, 4k + 2), k ∈ N, thenϕ(K,L) = ϕ(L,K) for every K,L ∈ Kn.

(c) If (i, n) = (2k + 1, 4k + 2), k ∈ N, then there exist an SO(n) invariantζ ∈ BVali,i and K,L ∈ Kn such that ζ(K,L) 6= ζ(L,K).

21

Proof : Since the cases i = 0 and i = n are trivial, we may assume that0 < i < n. Moreover, since O(n) × O(n) finite bivaluations are dense inBVali,i we may assume that ϕ ∈ BValfi,i, where 1 ≤ i ≤ n.

From Theorem 1 we deduce that the decomposition of the space Vali intoirreducible O(n) modules is multiplicity free, say

Vali =⊕γ∈R

Γγ,

where the sum ranges over some set R of equivalence classes of irreduciblerepresentations of O(n).

From Proposition 6.3, it follows that

BValf,O(n)i,i

∼= (Valfi ⊗Valfi )O(n) ∼=

⊕γ,δ∈R

(Γγ ⊗ Γδ)O(n).

Since, by Lemma 3.2, all representations of O(n) are self-dual, we have

(Γγ ⊗ Γδ)O(n) ∼= HomO(n)(Γγ,Γδ) ∼= HomO(n)(Γγ ⊗ Γδ,C).

Since Γγ and Γδ are irreducible, Schur’s lemma implies that

dim HomO(n)(Γγ,Γδ) =

1 if γ = δ0 if γ 6= δ.

Since all representations of O(n) are real, the space

HomO(n)(Γγ ⊗ Γγ,C) = (Sym2Γγ)O(n) ⊕ (Λ2Γγ)

O(n)

of O(n) invariant bilinear forms on Γγ coincides with (Sym2Γγ)O(n). Thus,

BValf,O(n)i,i

∼=⊕γ∈R

(Sym2Γγ)O(n)

which completes the proof of statement (a).

22

If n 6≡ 2 mod 4, then the proof of statement (b) is similar, since in thiscase all representations of SO(n) are also real, by Lemma 3.2. However, inthe case n ≡ 2 mod 4, more care is needed, since there are SO(n) moduleswhich are not real. By Lemma 3.2, an irreducible SO(n) module Γλ of highestweight λ = (λ1, . . . , λn/2) is real if and only if λn/2 = 0. If i 6= n/2, then, byTheorem 1, all irreducible SO(n) modules which enter Vali are of this form.Consequently, any SO(n) invariant bivaluation ϕ ∈ BVali,i is symmetric inthis case.

Finally let n ≡ 2 mod 4 and i = n/2. By Theorem 1, the dual irreducibleSO(n) modules Γ(2,...,2) and Γ(2,...,2,−2) both enter Vali with multiplicity one.

If φ1, . . . , φl is a basis of Γ(2,...,2) ⊆ Valfi and ψ1, . . . , ψl denotes the

corresponding dual basis in Γ(2,...,2,−2) ⊆ Valfi , then the image of∑l

k=1 φk⊗ψk

under the map ι defined in (6.1) clearly is a continuous SO(n) invariantbivaluation in BVali,i. However, it is not symmetric since the valuationsφ1, . . . , φl, ψ1, . . . , ψl are linearly independent.

7. Applications to geometric inequalities

As applications of Theorem 2, we present in this section several newgeometric inequalities involving SO(n) equivariant Minkowski valuations.Their proofs are based, on one hand, on the symmetry of bivaluations and,on the other hand, on techniques developed by Lutwak [37–41].

Lemma 7.1. If Φ ∈ MVal is SO(n) equivariant, then Φ is also O(n)equivariant.

Proof : Let CVal denote the vector space of all continuous and translationinvariant valuations with values in the space C(Sn−1) of continuous complexvalued functions on Sn−1. Note that any SO(n) equivariant Φ ∈ MValinduces an SO(n) equivariant Φ ∈ CVal, by Φ(K, ·) = h(ΦK, ·). Therefore,it is sufficient to show that any SO(n) equivariant valuation in CVal is O(n)equivariant.

Using arguments as in the proof of Proposition 6.3, it is easy to show thatCValf ∼= Valf ⊗ C(Sn−1)f as O(n)×O(n) modules. Consequently,

CValf,O(n) ∼= (Valf ⊗ C(Sn−1)f )O(n). (7.1)

23

It is well known that the decomposition of C(Sn−1) into irreducible SO(n)modules is given by

C(Sn−1) =⊕k≥0

Γ(k,0,...,0), (7.2)

where the spaces Γ(k,0,...,0) are precisely the spaces of spherical harmonics ofdegree k in dimension n. Moreover, the spaces Γ(k,0,...,0) are self-dual andO(n) invariant and, thus, (7.2) also represents the decomposition of C(Sn−1)into irreducible O(n) modules.

Let mk denote the (finite) multiplicity of the isomorphic copy of Γ(k,0,...,0)

in Val. From (7.1), Theorem 1 and an application of Schur’s lemma, weobtain

CValSO(n) =⊕

k

mk(Γ(k,0,...,0) ⊗ Γ(k,0,...,0))SO(n) = CValO(n).

Thus, any SO(n) equivariant valuation in CVal is also O(n) equivariant.

For K,L ∈ Kn and 0 ≤ i ≤ n−1, we write Wi(K,L) to denote the mixedvolume V (K, . . . ,K,B, . . . , B, L), where K appears n− 1− i times and theEuclidean unit ball B appears i times. The mixed volume Wi(K,K) willbe written as Wi(K) and is called the i-th quermassintegral of K. The i-thintrinsic volume Vi(K) of K is defined by

κn−iVi(K) =

(n

i

)Wn−i(K), (7.3)

where κn is the n-dimensional volume of the Euclidean unit ball in V .We will repeatedly make use of the following consequence of Theorem 2

and Lemma 7.1.

Corollary 7.2. If Φi ∈ MVali, 1 ≤ i ≤ n− 1, is SO(n) equivariant, then

Wn−1−i(K,ΦiL) = Wn−1−i(L,ΦiK)

for every K,L ∈ Kn.

Let Kno ⊆ Kn denote the set of convex bodies with non-empty interior.

One of the fundamental inequalities for mixed volumes is the (general)Minkowski inequality: If K,L ∈ Kn

o and 0 ≤ i ≤ n− 2, then

Wi(K,L)n−i ≥ Wi(K)n−i−1Wi(L), (7.4)

with equality if and only if K and L are homothetic.

24

Lemma 7.3. Let Φi ∈ MVali, 1 ≤ i ≤ n − 1, be SO(n) equivariant andnon-trivial, i.e. Φi(K) 6= 0 for some K ∈ Kn.

(a) There exists a constant r(Φi) > 0 such that for every K ∈ Kn,

Wn−1(ΦiK) = r(Φi)Wn−i(K).

(b) If K ∈ Kno , then

Wn−i(K)i+1 ≥ κin

r(Φi)i+1Wn−1−i(ΦiK).

If ΦiKno ⊆ Kn

o , then equality holds if and only if ΦiK is a ball.

Proof : Statement (a) follows from Hadwiger’s characterization theorem.From repeated application of Minkowski’s inequality (7.4) with L = B, weobtain the inequality

Wn−1(K)i+1 ≥ κinWn−1−i(K),

where, for K ∈ Kno , there is equality if and only if K is a ball. Taking ΦiK

instead of K and using (a), yields statement (b).

Special cases of Lemma 7.3 were previously obtained by Lutwak [37] (forΦi = Πi) and one of the authors [49] (for i = n− 1).

In order to proof Theorem 3, we need a further generalization of theBrunn–Minkowski inequality (1.3) (where the equality conditions are not yetknown): If 0 ≤ i ≤ n− 2, K,L,K1, . . . , Ki ∈ Kn and C = (K1, ..., Ki), then

Vi(K + L,C)1/(n−i) ≥ Vi(K,C)1/(n−i) + Vi(L,C)1/(n−i). (7.5)

Proof of Theorem 3 : Since translation invariant continuous Minkowskivaluations which are homogeneous of degree one are linear with respect toMinkowski addition (see e.g. [26]), the case i = 1 is a direct consequence ofinequality (1.3). Thus, we may assume that i ≥ 2.

By Corollary 7.2 and (7.5), we have for Q ∈ Kno ,

Wn−1−i(Q,Φi(K + L))1/i = Wn−1−i(K + L,ΦiQ)1/i

≥ Wn−1−i(K,ΦiQ)1/i +Wn−1−i(L,ΦiQ)1/i

= Wn−1−i(Q,ΦiK)1/i +Wn−1−i(Q,ΦiL)1/i.

25

It follows from Minkowski’s inequality (7.4), that

Wn−1−i(Q,ΦiK)i+1 ≥ Wn−1−i(Q)iWn−1−i(ΦiK), (7.6)

andWn−1−i(Q,ΦiL)i+1 ≥ Wn−1−i(Q)iWn−1−i(ΦiL). (7.7)

Thus, if we set Q = Φi(K+L) and use (7.3), we obtain the desired inequality

Wn−1−i(Φi(K + L))1/i(i+1) ≥ Wn−1−i(ΦiK)1/i(i+1) +Wn−1−i(ΦiL)1/i(i+1).

Suppose now that equality holds and that ΦiKno ⊆ Kn

o . By Theorem 1′,applied to the standard representation V = Γ(1,0,...,0), the Steiner point ofΦiK is the origin for every K ∈ Kn. Thus, we can deduce from the equalityconditions of (7.6) and (7.7), that there exist λ1, λ2 > 0 such that

ΦiK = λ1Φi(K + L) and ΦiL = λ2Φi(K + L) (7.8)

andλ

1/i1 + λ

1/i2 = 1.

Using Lemma 7.3 (a) and (7.8) we get

Wn−i(K + L)1/i = Wn−i(K)1/i +Wn−i(L)1/i,

which, by (1.3), implies that K and L are homothetic.

The major open problem concerning the rigid motion invariant quantitiesWn−1−i(ΦiK) is how to estimate them from below in terms of Wn−1−i(K).A standard method of proof for isoperimetric problems of this kind wasintroduced by Lutwak [38] and is now known as the class reduction technique.Our last result shows how Corollary 7.2 allows for applications of the classreduction technique to the functionals Wn−1−i(ΦiK), K ∈ Kn.

In the following we use Φ2iK to denote ΦiΦiK.

Theorem 7.4. Let Φi ∈ MVali, 1 ≤ i ≤ n − 1, be SO(n) equivariant andsuppose that ΦiKn

o ⊆ Kno . If K ∈ Kn

o , then

Wn−1−i(ΦiK)

Wn−1−i(K)i≥ Wn−1−i(Φ

2iK)

Wn−1−i(ΦiK)i,

with equality if and only if K and Φ2iK are homothetic.

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Proof : Let K,Q ∈ Kno . From Corollary 7.2 and the Minkowski inequality

(7.4), we obtain

Wn−1−i(Q,ΦiK)i+1 = Wn−1−i(K,ΦiQ)i+1 ≥ Wn−1−i(K)iWn−1−i(ΦiQ),

with equality if and only if K and ΦiQ are homothetic. Taking Q = ΦiK,yields

Wn−1−i(ΦiK)i+1 ≥ Wn−1−i(K)iWn−1−i(Φ2iK),

with equality if and only if K and Φ2iK are homothetic.

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Tel Aviv University, IsraelE-mail address: semyon@post.tau.ac.il

Goethe-Universitat Frankfurt am Main, GermanyE-mail address: bernig@math.uni-frankfurt.de

Vienna University of Technology, AustriaE-mail address: franz.schuster@tuwien.ac.at

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